TITLE: Geometry problem, finding missing angles
QUESTION [9 upvotes]: One of my students showed me a problem that she says is similar to what they would do in high school in her home country (which i am attaching here.
. The goal of the problem is to find the measure of $\angle DEC$ using the given angle measures provided. I've tried working on this to see what other angle measures I could deduce, and I'm including that here  This is where I'm stuck. I've tried:

Labeled one unknown angle as x and determined all other unknown angles in terms of $x$, but it's fully consistent and nothing seems to simplify to indicate what $x$ is.
Drawn in lines parallel to the sides through various points, and use what I know about parallel lines cut by transversals, but it doesn't seem to get me any closer to the target.

I suspect I might need to draw in some additional line or extend the diagram in some way but I can't figure out what. Any help would be greatly appreciated. Thanks!

REPLY [3 votes]: Extend $BC$ to $F$ such that $∠BFA = ∠DEB$.
Then, $AEDF$ is a cyclic quadrilateral.
$∠DAF = 180° - 30° - 60° = 90°$
Therefore, the center of the circle AEDF should lie on DF. Let O be the center.
$∠AOC = 2∠AFD = 60° = ∠ODG$
$AO = OD$
$∠OAC = ∠OAD - ∠CAD = 20° = 2∠DAE = ∠DOG$
Then, $△OAC≅△DOG$.
$∠DEG = 180° - 20° - ∠DGE = 160° - 60° -20° = 80° = ∠DGE$
Hence, $OC = DE$. Moreover, $∠EOC = 2(10°) = ∠ADE$ and $OE = DA$.
Then, $△EOC≅△ADE$.
$∠DEC = 180° - 30° - ∠CEO - ∠OEA = 150° - ∠EAD - 70° = 70°$